Use the graph below, which shows the profit, *y*, in thousands of dollars, of a company in a given year, *t*, where *t* represents the number of years since 1980.
Find the linear function *y*, where *y* depends on *t*, the number of years since 1980.

To find the linear function \( y(t) \) that represents the profit \( y \) in thousands of dollars as a function of time \( t \), where \( t \) is the number of years since 1980, we need to follow these steps:

### 1. Identify Two Points on the Line
Since the function is linear, we need two points on the line to determine its equation. Let's assume the graph provides us with two points: \( (t_1, y_1) \) and \( (t_2, y_2) \).

### 2. Calculate the Slope \( m \)
The slope \( m \) of the line can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{t_2 - t_1}
\]
This slope represents the change in profit per year.

### 3. Write the Equation of the Line
Once we have the slope \( m \), we can write the equation of the line in point-slope form, which can be converted to the slope-intercept form:
\[
y = m \cdot t + b
\]
where \( b \) is the y-intercept, representing the profit at \( t = 0 \) (which corresponds to the year 1980).

### 4. Determine the y-Intercept \( b \)
We can find \( b \) by substituting one of the points \( (t_1, y_1) \) into the equation:
\[
y_1 = m \cdot t_1 + b
\]
Solve for \( b \):
\[
b = y_1 - m \cdot t_1
\]

### 5. Write the Final Equation
Substitute the values of \( m \) and \( b \) into the linear function:
\[
y(t) = m \cdot t + b
\]

**Example Calculation** (if specific points were given):
If we assume two points from the graph are \( (2, 10) \) and \( (5, 25) \), we would calculate the slope:
\[
m = \frac{25 - 10}{5 - 2} = \frac{15}{3} = 5
\]
Then, using \( (2, 10) \) to find \( b \):
\[
10 = 5 \cdot 2 + b \Rightarrow b = 10 - 10 = 0
\]
So, the linear function would be:
\[
y(t) = 5t
\]

If you can provide specific points from the graph or an image, I can calculate the exact linear function for you.

Would you like any details on this explanation or have any specific questions?

### Related Questions:
1. How do you interpret the slope in the context of this problem?
2. What would happen to the linear function if the company had a profit of zero at some year after 1980?
3. How would you find the year in which the profit was a specific value?
4. What if the graph was non-linear—how would that change the approach?
5. How could external factors affect the accuracy of this linear model over time?

**Tip:** Always check if the relationship is truly linear by plotting the points or analyzing more than two points to ensure consistency.

Use the graph below, which shows the profit, y, in thousands of dollars, of a company in a given year, t, where t represents the number of years since 1980. Find the linear function y, where y depends on t, the number of years since 1980.

Learn how to determine the linear function representing a company's profit over time, given a graph with data starting from 1980. This problem covers algebraic concepts such as slope, point-slope form, and slope-intercept form. Suitable for students in Grades 9-10.

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